I have a context free grammar (CFG). The initial symbol is `A1`. The terminal symbols are just integers and the non-terminal symbols start with `A`. I also have a target sequence of terminal symbols which needs to be produced by the grammar. However, the CFG production rules have had some terminal symbols removed which may now render it impossible to produce the target sequence. Please attempt to repair the CFG so that the sequence can be produced. The only operations which can be made to repair the CFG are the insertion of terminal symbols anywhere in existing production rules. Deleting or reordering symbols in existing production rules and creating new production rules is prohibited. Please try minimize the total number of terminal symbols inserted into the production rules so that the CFG can produce the target sequence. Reason about the problem however you wish and write out whatever working out you want. When you are finished please write a new line with `Updated production rules:`, and then write a new line for each of the updated production rules (including `rule_<n>:`), no need to rewrite the production rules which haven't changed. Do not write anything after the updated production rules.


"""
There are 22 terminals, ranging from 1 to 22.
Each terminal symbol occurs in the target sequence at least once.

There are 38 non-terminal symbols, ranging from A1 to A38.

Production rules:
rule_1: A1 -> A34 A29 A24 A18 A2
rule_2: A2 -> 1
rule_3: A2 -> 1 A6 A9 3
rule_4: A2 -> A7 A6 3
rule_5: A2 -> A10 A6
rule_6: A2 -> A14 A6 4
rule_7: A3 -> 1
rule_8: A4 -> 1
rule_9: A5 -> 1
rule_10: A5 -> 2
rule_11: A6 -> 1
rule_12: A7 -> 1
rule_13: A7 -> 5
rule_14: A8 -> 1
rule_15: A8 -> 6
rule_16: A9 -> A7 4
rule_17: A9 -> A8
rule_18: A10 -> A11
rule_19: A10 -> A13
rule_20: A11 -> A12
rule_21: A11 -> A12 7
rule_22: A12 -> 1
rule_23: A13 -> A5
rule_24: A14 -> A15
rule_25: A14 -> A17
rule_26: A15 -> A16 1
rule_27: A16 -> 1
rule_28: A17 -> A7 1
rule_29: A18 -> 1
rule_30: A18 -> A4 A19
rule_31: A18 -> A6 A21 A23
rule_32: A18 -> A7 A21 9
rule_33: A18 -> A8 A21 9
rule_34: A19 -> 1
rule_35: A19 -> A3 A20
rule_36: A20 -> A5
rule_37: A21 -> 1
rule_38: A22 -> A8 11
rule_39: A23 -> A8 10
rule_40: A23 -> 12
rule_41: A24 -> 1
rule_42: A24 -> A8 A25 13
rule_43: A24 -> A21 A25 A26
rule_44: A24 -> A27 A25 14 13
rule_45: A25 -> 1
rule_46: A25 -> A6 A22
rule_47: A26 -> A7
rule_48: A26 -> A8 14
rule_49: A27 -> A28
rule_50: A28 -> A5 A19 15
rule_51: A28 -> A5 A19 16
rule_52: A29 -> 1
rule_53: A29 -> A31 A30 A33 17
rule_54: A30 -> 1
rule_55: A31 -> 1
rule_56: A31 -> A19 A32
rule_57: A32 -> A4
rule_58: A32 -> A5
rule_59: A33 -> A4
rule_60: A33 -> A5
rule_61: A33 -> 18
rule_62: A34 -> 1
rule_63: A34 -> 1 A35 A36 19
rule_64: A34 -> A5 A35 20
rule_65: A34 -> A37 A35 20 19
rule_66: A35 -> 1
rule_67: A36 -> A4
rule_68: A36 -> A5
rule_69: A37 -> A38
rule_70: A38 -> A8 1 21 22

Target sequence:
1 1 21 22 2 1 20 19 1 16 1 15 1 18 17 6 1 12 1 11 14 13 1 1 10 9 1 1 7 8 5 1 4 3
"""